Method for constructing co-rotating, contiguous bodies and computer program product for carrying out said method

Abstract

The invention relates to a method of constructing elements which wipe each other during corotation about two parallel axes in such a manner that they constantly touch each other at at least one point.

The invention claimed is: 
     
       1. A method of generating two elements K 1  and K 2  that co-rotate about two parallel axes that are at a distance a from each other, the method comprising the steps of:
 providing a profile of element K 1  having n number of arcs; 
 merging the arcs into each other tangentially at their starting and end points such that 
 the arcs forming a convex profile, 
 wherein a kink in the profile is represented by an arc i with a radius r_i=0 and an angle α_i, and wherein a size of the angle α_i is the same as that of the angle at which the tangents on the arcs on adjacent to arc i intersect each other at the centre point of arc i; and 
 forming a profile of element K 2  from the profile of element K 1  by generating, for each arc of the profile of element K 1 , a corresponding arc of element K 2 , 
 wherein the two elements K 1  and K 2  constantly touch each other at at least one point during the co-rotation. 
 
     
     
       2. The method according to  claim 1 , wherein
 the angles of corresponding arcs of element K 2  are identical in size, 
 the sum of the radii of corresponding arcs equals the centre distance a, 
 each one of the connecting lines between the centre point of an arc of element K 1  and its end points is parallel to each one of the connecting lines between the centre point of the corresponding arc of element K 2  and its end points, 
 those directions in which the end points of an arc of element K 1  lie 
 starting from the centre point of said arc are in each case opposite those directions in which the end points of the corresponding arc of element K 2  lie starting from the centre point of said arc of element K 2 , 
 the distance between the centre point of an arc of element K 1  and the centre point of a corresponding arc of element K 2  equals the centre distance, 
 the connecting line between the centre point of an arc of element K 1  and the centre point of the corresponding arc of element K 2  is parallel to the connecting line between the point of rotation of element K 1  and the point of rotation of element K 2 , and 
 the direction in which the centre point of an arc of element K 1  would have to be shifted in order to fit exactly over the centre point of the corresponding arc of element K 2  is the same as that in which the point of rotation of element K 1  must be shifted in order to fit exactly over the point of rotation of element K 2 . 
 
     
     
       3. The method according to  claim 1 , wherein
 a number of arcs n is selected for forming the profile of element K 1 , wherein n is an integer which is greater than or equal to 1, 
 an outer radius ra is selected, wherein ra can be greater than 0 (ra>0) and smaller than or equal to the centre distance (ra≦a), 
 an inner radius ri is selected, wherein ri can be greater than or equal to 0 (n≧0) and smaller than or equal to ra (ri≦ra), 
 n arcs of element K 1  are arranged clockwise or anti-clockwise about the axis of rotation of element K 1  in such a manner that the sizes of n−1 arcs are determined by the selectable angles α_ 1 , α_ 2 , . . . , α_(n−1) and the selectable radii r_ 1 , r_ 2 , . . . , r_(n−1), wherein the angles are, in terms of radian measurement, greater than or equal to 0 and smaller than or equal to 2π and the radii are greater than or equal to 0 and smaller than or equal to the centre distance a; 
 the angle α_n of a last arc is determined by the fact that the sum of n angles of the n arcs is, in terms of radian measurement, 2π; 
 the radius r_n of a last arc is determined by the fact that this last arc closes the profile; 
 all of the arcs merge into each other tangentially to form a convex profile, 
 an arc with a radius of 0 is treated preferably in the same way as an arc with a radius of eps, which is a very small positive real number which tends towards 0 (eps<<1, eps→0); 
 all of the arcs are located within or on the boundary of an annulus which has an outer radius ra and an inner radius ri and whose centre point is located on the point of rotation of element K 1 ; 
 at least one of the arcs is in contact with the outer radius ra; and 
 at least one of the arcs is in contact with the inner radius ri. 
 
     
     
       4. The method according to  claim 3 , wherein the arcs i′ of the profile of element K 2  are based on the arcs i of the profile of element K 1  such that
 the number of arcs n′ is the same as n; 
 i and i′ are integers which together represent all values in the range from 1 to the number of arcs n and n′ respectively (i′=i), 
 the following applies to the angles of arcs i′: α_ 1 ′=α_ 1 ; α_ 2 ′=α_ 2 ; . . . ; α_n′=α_n; 
 the following applies to the radii of arcs i′: r_ 1 ′=a−r_ 1 ; r_ 2 ′=a−r_ 2 ; . . . r_n′=a−r_n; and 
 the distance between the centre point of the i′nth arc of the profile of element K 2  and the centre point of the inth arc of the profile of element K 1  is equal to the centre distance a, 
 the distance between the centre point of the i′nth arc of the profile of element K 2  and the point of rotation of element K 2  corresponds to the distance between the centre point of the inth arc of the profile of element K 1  and the point of rotation of element K 1 , 
 the connecting line between the centre point of the i′nth arc of the profile of element K 2  and the centre point of the inth arc of the profile of element K 1  is a line parallel to the connecting line between the point of rotation of element K 2  and the point of rotation of element K 1 , and 
 a starting point of the i′nth arc of the profile of element K 2  lies in an opposite direction, in relation to the centre point of the i′nth arc of the profile of element K 2 , to that in which a starting point of the inth arc of the profile of element K 1  lies in relation to the centre point of the inth arc of the profile of element K 1 . 
 
     
     
       5. The method according to  claim 1 , wherein, when using a Cartesian coordinate system with the point of rotation of the profile of element K 1  at the origin (x=0, y=0) and the point of rotation of element K 2  at the point having the coordinates x=A=1, y=0 and when using dimensionless parameters, the profile of element K 1  is formed by the following steps:
 a number of arcs n is selected for forming the profile of element K 1 , wherein n is an integer which is greater than or equal to 1; 
 an outer radius RA is selected which is greater than 0 (RA>0) and smaller than or equal to the centre distance (RA≦1); 
 an inner radius RI is selected which is greater than or equal to 0 (RI≧0) and smaller than or equal to RA (RI≦RA); and 
 arcs i of element K 1  are arranged clockwise or anticlockwise around the axis of rotation of element K 1 , wherein i is an index which represents the integers in the range from 1 to n, such that
 the sum of the angles α_i of all of the arcs is 2π; 
 the radius R_i of each individual arc is greater than or equal to 0 and smaller than or equal to 1; 
 the starting and centre points of a first arc are placed on the x axis, the starting point being placed in the region between x=RI and x=RA and the x coordinate of the centre point being smaller than or equal to the x coordinate of the starting point; 
 where i<n, the end point of the inth arc is at the same time the starting point of the (i+1)nth arc; 
 where i=n, the end point of the inth arc is at the same time the starting point of the first arc; 
 each arc merges tangentially into the next arc, wherein an arc with R_i=0 is treated in the same way as an arc with R_i=eps, wherein eps is a very small positive real number which tends towards 0 (eps<<1, eps→0); 
 at no point of the profile is the distance from the point of rotation greater than the outer radius RA; 
 at at least one point on the profile the distance from the point of rotation is equal to the outer radius RA; 
 at no point of the profile is the distance from the point of rotation smaller than the inner radius RI; 
 at at least one point on the profile the distance from the point of rotation is equal to the inner radius RI and 
 the profile is convex. 
 
 
     
     
       6. The method according to  claim 5 , wherein the profile of element K 2  is based on the profile of element K 1  as follows:
 n′=n 
 i′ is an index which represents all numbers from 1 to n′ 
 α_i′=α_i where i=i′ and 
 R_i′+R_i=1 where i=i′, 
 the profile consists of n′ arcs which are arranged in the same clock direction as the arcs of the profile of element K 1 , 
 the starting point of the first arc of the profile of element K 2  is at the same time the starting point of the first arc of the profile of element K 1  and the centre point of the first arc of the profile of element K 2  is located on the x axis, the x coordinate of the centre point being greater than or equal to the x coordinate of the starting point, 
 where i′<n′ the end point of the i′nth arc is at the same time the starting point of the (i+1)′nth arc, 
 where i′=n′ the end point of the i′nth arc is at the same time the starting point of the 1′nth arc, 
 each of the arcs merges tangentially into the next arc, an arc with R_i′=0 being treated as an arc with R 13  i′=eps, wherein eps is a very small positive real number which tends towards 0 (eps<<1, eps→0), and 
 the profile is convex. 
 
     
     
       7. The method according to  claim 1 , wherein the profile of element K 1  is formed such that a flight number z is selected, wherein z is an integer which is greater than or equal to 1;
 the n number of arcs is selected such that it is an integer multiple p of 4*z; 
 the profile is subdivided into 2*z sections wherein each section is bounded by two straight lines which form an angle to each other, in terms of radian measurement, of π/z and which intersect each other at the point of rotation of the profile, wherein these two straight lines are referred to as section boundaries; 
 each of these 2*z sections is subdivided into a first and a second part; 
 the first part of a section is composed of p arcs which are numbered in ascending or descending order; 
 angles α_ 1 , . . . , α_p of the p arcs are selected so as to give a sum of these angles of π/(2*z), wherein the angles are, in terms of radian measurement, greater than or equal to 0 and smaller than or equal to π/(2*z); 
 the second part of a section is composed of p′ arcs which are numbered in reverse order to the arcs of the first part of a section, wherein p′ is an integer which is the same as p; 
 angles α_p′, . . . , α_ 1 ′ of the p′ arcs are formed in such a manner that the angle α_j′ of the j′th arc of the second part of a section is the same as the angle α_j of the jth arc of the first part of a section, wherein j and j′ are integers which together represent all values in the range from 1 to the number of arcs p and p′ respectively (α_ 1 ′=α_ 1 , . . . , α_p′=α_p); 
 the sum of the radius r_j′ of the j′th arc of the second part of a section and the radius r_j of the jth arc of the first part of a section is equal to the centre distance a, wherein j and j′ are integers which together represent all values in the range from 1 to the number of arcs p and p′ respectively (r_ 1 ′+r_ 1 =a, . . . , r_p′+r_p=a); 
 a centre point and a starting point of the arc with which the profile begins in the first part of a section are positioned on one of the section boundaries of this section, depending on whether the arcs are arranged in a clockwise or an anticlockwise direction; and 
 an end point of the arc with which the profile ends in the first part of a section touches a straight line FP at one point, the straight line FP being vertical on the angle bisector of the two section boundaries of this section and at a distance from the point of rotation of the profile in the direction of this section which is equal to half the centre distance, the angle bisector passing, in the same way as the section boundaries, through the point of rotation of the profile. 
 
     
     
       8. The method according to  claim 7 , wherein one section in one of the 2*z sections of the generating screw profile is predefined and the remaining sections of the screw profile are generated by continuous mirroring of the predefined screw profile section at the section boundaries. 
     
     
       9. The method according to  claim 8 , wherein the case of odd flight numbers the profile of element K 2  is identical to the profile of element K 1  and in the case of even flight numbers the profile of element K 2  is obtained by rotating the profile of element K 1  through an angle π/z. 
     
     
       10. The method according to  claim 1 , wherein the profiles extend in an axial direction in the shape of a screw, the elements generated in this manner being of left-hand or right-hand direction and having a pitch, standardized on the centre distance, which is in the range from 0.1 to 10 and a length, standardized on the centre distance, which is in the range from 0.1 to 10. 
     
     
       11. The method according to  claim 1 , wherein the profiles extend stepwise in a linear fashion in an axial direction and the length of the elements, standardized on the centre distance, is in the range from 0.05 to 10. 
     
     
       12. The method according to  claim 1 , wherein a transitional element is generated by forming a continuous transition from the size(s) and/or position(s) of one or more arcs of a first profile to the size(s) and/or position(s) of one or more arcs of a second profile, and wherein the transitional element is in a right-hand or left-hand direction and has a pitch, standardized on the centre distance, in the range from 0.1 to 10 and an element length, standardized on the centre distance, in the range from 0.1 to 10. 
     
     
       13. A computer program product with program code means stored on a computer-readable data carrier for carrying out the method according to  claim 1  upon executing the computer program product on a computer.